We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $$w\in S_n$$ w ∈ S n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.

We compute projective dimension of translated simple modules in the regular block of the Bernstein–Gelfand–Gelfand category $\mathcal{O}$ in terms of Kazhdan–Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan Kåhrström. We also derive some inequalities, in terms of Lusztig’s $\textbf{a}$-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we prove that Kostant’s problem is equivalent to a homological problem of decomposing translated simple modules in $\mathcal O$. This gives a conjectural answer to Kostant’s problem in terms of the Kazhdan–Lusztig basis and addresses yet another conjecture by Johan Kåhrström.

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